Complex bordism
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1 Introduction
Complex bordism (also known as unitary bordism) is the bordism theory of stably complex manifolds. It is one of the most important theories of bordism with additional structure, or B-bordism.
The theory of complex bordism is much richer than its unoriented analogue, and at the same time is not as complicated as oriented bordism or other bordism theories with additional structure (B-bordism). Thanks to this, the complex cobordism theory has found the most striking and important applications in algebraic topology and beyond. Many of these applications, including the formal group techniques and Adams-Novikov spectral sequence were outlined in the pioneering work [Novikov1967].
2 Stably complex structures
A direct attempt to define the bordism relation on complex manifolds fails because the manifold is odd-dimensional and therefore cannot be complex. In order to work with complex manifolds in the bordism theory, one needs to weaken the notion of a complex structure. This leads directly to considering stably complex (also known as weakly almost complex, stably almost complex or quasicomplex) manifolds.
Let denote the tangent bundle of , and the product vector bundle over . A tangential stably complex structure on is determined by a choice of an isomorphism
between the "stable" tangent bundle and a complex vector bundle over . Some of the choices of such isomorphisms are deemed to be equivalent, i.e. determining the same stably complex structures (see details in Chapters II and VII of [Stong1968]). In particular, two stably complex structures are equivalent if they differ by a trivial complex summand. A normal stably complex structure on is determined by a choice of a complex bundle structure on the normal bundle of an embedding . Tangential and normal stably complex structures on determine each other by means of the canonical isomorphism . We therefore may restrict our attention to tangential structures only.
A stably complex manifold is a pair consisting of a manifold and a stably complex structure on it. This is a generalisation of a complex and almost complex manifold (where the latter means a manifold with a choice of a complex structure on , i.e. a stably complex structure with ).
Example 2.1. Let . The standard complex structure on is equivalent to a stably complex structure determined by the isomorphism
where is the Hopf line bundle. On the other hand, the isomorphism
determines a trivial stably complex structure on .
3 Definition of bordism and cobordism
The bordism relation can be defined between stably complex manifolds. Like the case of unoriented bordism, the set of bordism classes of stably complex manifolds of dimension is an Abelian group with respect to the disjoint union. This group is called the -dimensional complex bordism group and denoted . The zero element is represented by the bordism class of any manifold which bounds and whose stable tangent bundle is trivial (and therefore isomorphic to a product complex vector bundle ). The sphere provides an example of such a manifold. The opposite element to the bordism class in the group may be represented by the same manifold with the stably complex structure determined by the isomorphism
where is given by .
An abbreviated notation for the complex bordism class will be used whenever the stably complex structure is clear from the context.
The complex bordism group and cobordism group of a space may also be defined geometrically, at least for the case when is a manifold. This can be done along the lines suggested by [Quillen1971a] and [Dold1978] by considering special "stably complex" maps of manifolds to . However, nowadays the homotopical approach to bordism has taken over, and the (co)bordism groups are usually defined using the Pontrjagin-Thom construction similarly to the unoriented case:
where is the Thom space of the universal complex -plane bundle . These groups are -modules and give rise to a multiplicative (co)homology theory. In particular, is a graded ring.
The graded ring with is called the complex cobordism ring; it has nontrivial elements only in nonpositively graded components.
4 Geometric cobordisms
There is one important case when certain cobordism classes can be represented very explicitly by maps of manifolds.
For any cell complex the cohomology group can be identified with the set of homotopy classes of maps into . Since , every element also determines a cobordism class . The elements of obtained in this way are called geometric cobordisms of . We therefore may view as a subset in , however the group operation in is not obtained by restricting the group operation in (see Formal group laws and genera for the relationship between the two operations).
When is a manifold, geometric cobordisms may be described by submanifolds of codimension 2 with a fixed complex structure on the normal bundle.
Indeed, every corresponds to a homotopy class of maps . The image is contained in some , and we may assume that is transversal to a certain hyperplane . Then is a codimension 2 submanifold in whose normal bundle acquires a complex structure by restriction of the complex structure on the normal bundle of . Changing the map within its homotopy class does not affect the bordism class of embedding .
Conversely, assume given a submanifold of codimension 2 whose normal bundle is endowed with a complex structure. Then the composition
of the Pontrjagin-Thom collapse map and the map of Thom spaces corresponding the the classifying map of defines and element , and therefore a geometric cobordism.
If is an oriented manifold, then a choice of complex structure on the normal bundle of a codimension 2 embedding is equivalent to orienting . The image of the fundamental class of in the homology of is Poincare dual to .
5 Structure results
Complex bordism ring is described as follows.
Theorem 5.1.
- is a polynomial ring over generated by the bordism classes of complex projective spaces , .
- Two stably complex manifolds are bordant if and only if they have identical sets of Chern characteristic numbers.
- is a polynomial ring over with one generator in every even dimension , where .
Part 1 can be proved by the methods of [Thom1954]. Part 2 follows from the results of [Milnor1960] and [Novikov1960]. Part 3 is the most difficult one; it was done by [Novikov1960] using Adams spectral sequence and structure theory of Hopf algebras (see also [Novikov1962] for a more detailed account) and Milnor (unpublished, but see [Thom1995]) in 1960. Another more geometric proof was given by [Stong1965], see also [Stong1968].
6 Multiplicative generators
6.1 Preliminaries: characteristic number sn
To describe a set of multiplicative generators for the ring we shall need a special characteristic class of complex vector bundles. Let be a complex -plane bundle over a manifold~. Write formally its total Chern class as follows:
so that is the th elementary symmetric function in formal indeterminates. These indeterminates acquire a geometric meaning if is a sum of line bundles; then , . Consider the polynomial
and express it via the elementary symmetric functions:
Substituting the Chern classes for the elementary symmetric functions we obtain a certain characteristic class of :
This characteristic class plays an important role in detecting the polynomial generators of the complex bordism ring, because of the following properties (which follow immediately from the definition).
Proposition 6.1.
- for .
- .
Given a stably complex manifold of dimension , define its characteristic number
where is the complex bundle from the definition of stably complex structure, and the fundamental homology class.
Corollary 6.2. If a bordism class decomposes as where and , then .
It follows that the characteristic number vanishes on decomposable elements of . It also detects indecomposables that may be chosen as polynomial generators. In fact, the following result is a byproduct of the calculation of :
Theorem 6.3. A bordism class may be chosen as a polynomial generator of the ring if and only if
6.2 Milnor hypersurfaces Hij
A universal description of connected manifolds representing the polynomial generators is unknown. Still, there is a particularly nice family of manifolds whose bordism classes generate the whole ring . This family is superfluous though, so there are algebraic relations between their bordism classes.
Fix a pair of integers and consider the product . Its algebraic subvariety
is called the Milnor hypersurface. Note that .
Denote by and the projections onto the first and second factors respectively, and by the Hopf line bundle over a complex projective space; then is the hyperplane section bundle. We have
where , .
Proposition 6.4. The geometric cobordism in corresponding to the element is represented by the submanifold . In particular, the image of the fundamental class in is Poincare dual to .
See the proof.
Lemma 6.5. We have
See the proof.
Theorem 6.6. The bordism classes multiplicatively generate the complex bordism ring .
Proof. This follows from the fact that
and the previous Lemma.
Example 6.7.
- ;
- , generated by a point;
- , generated by , as and ;
- , generated by and , as and ;
- cannot be taken as the polynomial generator , since , while . One may take as the bordism class .
The previous theorem about the multiplicative generators for has the following important specification.
Theorem 6.8 (Milnor). Every bordism class with contains a nonsingular algebraic variety (not necessarily connected).
(The Milnor hypersufaces are algebraic, but one also needs to represent by algebraic varieties!) For the proof see Chapter 7 of [Stong1968].
The following question is still open, even in complex dimension 2.
Theorem 6.9 (Hirzebruch). Describe the set of bordism classes in containing connected nonsingular algebraic varieties.
Example 6.10. Every class contains a nonsingular algebraic variety, namely, a disjoint union of copies of for and a Riemannian surface of genus for . Connected algebraic varieties are only contained in the bordism classes with .
6.3 Toric generators Bij and quasitoric representatives in cobordism classes
7 Adams-Novikov spectral sequence
The main references here are [Novikov1967] and [Ravenel1986]
8 References
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- [Milnor1960] J. Milnor, On the cobordism ring and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR0119209 (22 #9975) Zbl 0095.16702
- [Novikov1960] S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Math. Dokl. 1 (1960), 717–720. MR0121815 (22 #12545) Zbl 0094.35902
- [Novikov1962] S. P. Novikov, Homotopy properties of Thom complexes, Mat. Sb. (N.S.) 57 (99) (1962), 407–442. MR0157381 (28 #615) Zbl 0193.51801
- [Novikov1967] S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Math. USSR, Izv. 1, (1967) 827–913. MR0221509 (36 #4561) Zbl 0176.52401
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- [Ravenel1986] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press Inc., Orlando, FL, 1986. MR860042 (87j:55003) Zbl 1073.55001
- [Stong1965] R. E. Stong, Relations among characteristic numbers. I, Topology 4 (1965), 267–281. MR0192515 (33 #740) Zbl 0136.20503
- [Stong1968] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968. MR0248858 (40 #2108) Zbl 0277.57010
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
- [Thom1995] R. Thom, Travaux de Milnor sur le cobordisme, Séminaire Bourbaki, Vol. 5, Exp. No. 180, Soc. Math. France, Paris, (1995), 169–177. MR1603465 Zbl 0116.40402
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